Simulation of two-phase flows in a heterogeneous porous medium may require considerable computer resources, in particular when the numerical model of the medium considered is greatly detailed. This is notably the case in reservoir engineering, in the petroleum sphere. In order to perform a flow simulation with reasonable means, a reduced description of the reservoir is necessary. One technique consists in aggregating the grid cells of the numerical model so as to obtain a rougher grid, made up of cells with mean effective values. This technique, referred to as pseudo-function technique, was initially proposed for stratified reservoirs so as to aggregate the cells in the vertical direction by:                Coats, K. H., Nielsen, R. L. and Terhune, M. H. “Simulation of three-dimensional, two-phase flow in oil and gas reservoirs”. SPE 1961, 1967.        
This method, taken up and extended by many authors, is based on capillary vertical equilibrium or gravity equilibrium hypotheses where the capillary or gravity forces are predominant in relation to the effects due to viscosity, and the distribution of the saturations in the reservoir can be known without requiring a simulation on the precise grid (fine simulation). Mean values can be readily obtained therefrom in the vertical direction. A single mean-value layer may be eventually sufficient to describe the evolution of the saturations in the reservoir.
Another method allowing mean values to be obtained is proposed by:                Hearn, C. L. “Simulation of stratified waterflooding by pseudo-relative permeability curve”. Journal of Petroleum Technology, pp. 805-813, July 1971.        
It is based on the hypothesis of vertical equilibrium due to the viscous effects. In this case, the viscous readjustments in reservoirs are very fast in relation to the saturation variations, so that mean values can always be calculated without using fine simulation, because the fluids are propagated at constant velocity in each layer. The Hearn method, later extended, among others, by:                Simon, A. D. and Koederitz, L. F. “An improved method for the determination of pseudo-relative permeability data for stratified systems”. SPE 10975, 1982,is based on an iterative calculation leading to arrange the reservoir layers according to the rate of propagation of the fluids in said layers. It can be shown that this iterative calculation may not converge. We then have a stationary front in at least two reservoir layers. The authors then suggest to take only one mean property layer for these particular zones. A theoretical study of the vertical equilibrium and of its implications can be found in:        Yortsos, Y. C. “Analytical studies for processes at vertical equilibrium”. SPE 26022, 1992.        
When the capillarity or the gravity cannot be considered to be predominant on the viscous effects, but the viscous vertical equilibrium cannot be reached, the methods proposed so far cannot do without a fine simulation on all or part of the reservoir to calculate the mean properties that can be assigned to the rough blocks. We then refer to dynamic pseudo-functions, which were notably introduced by:                Kyte, J. R. and Berry, D. W. “New pseudo functions to control numerical dispersion”. SPE 5105, 1975.        
These dynamic methods however pose many theoretical and practical problems. Reviews of these methods and of associated problems can be found in the following publications:                Archer, R. “Pseudo function generation”, Master of Science thesis, Department of Petroleum Engineering of Stanford University, 1996,        Ahmadi, A. “Utilisation des proprietes equivalentes dans les modeles de réservoir: cas des écoulements diphasiques incompressibles”. These de Doctorat, Université de Bordeaux I, 1992,        Barker, J. W. and Thibeau, S. “A critical review of the use of pseudo relative permeabilities for upscaling”. SPE 35491, 1996.        
Dynamic methods generally involve systematic and uniform aggregation of the grid cells. In order to take account of the local influences of heterogeneities on the flow, non-uniform aggregation methods were proposed, notably by:                Darman, N. H. and Durlofsky, L. J. “Upscaling immiscible gas displacements: Quantitative Use of Fine Grid Flow Data in Grid Coarsening Schemes”. SPE 59452, 2000.        
These methods have the advantage of aggregating only preferential zones and leave the parts of the reservoir with more complex hydrodynamic features in the initial state. They however cannot do without a fine simulation.
Methods allowing non-uniform aggregation of the strata of a reservoir without using a detailed simulation have been proposed. Entirely static methods such as:                Li, D. And Beckner, B. “Optimal uplayering for scaleup of multimillion-cell geologic models”, SPE 62927, 2000can be mentioned, or methods based on the results of a single-phase flow simulation (which is much faster than a two-phase flow simulation), such as:        Stern, D. And Dawson, A. G. “A technique for generating reservoir simulation grids to preserve geologic heterogeneity”, SPE 51942, 1999.        
Such methods, by definition, do not take account of the dynamic and viscous effects of two-phase flows.
The evolution of the front in the reservoir during the flow is considerably influenced by the viscous coupling between the pressure field and the saturation field. In particular, when the fluid injected is less viscous and consequently more mobile at the front level than the fluid in place, viscous instabilities will always favour the flow of fluids in the most permeable layers. The breakthrough time through these layers is much faster than in the rest of the reservoir. On the other hand, if the fluid injected is less mobile, the viscous coupling can slow it down in the initially faster layers, thus compensating for the permeability differences due to the stratification. A stationary front then appears.
If in at least part of the reservoir such a stationary front exists, everything goes on as if the flow encountered a single layer in the hydrodynamic sense. One can then reasonably think that a single layer with mean properties in this zone is sufficient for modelling upon numerical simulation of the flow. This is of major importance when switching from the geological reservoir model with all the layers to the simulation model, for which only the most influential zones as regards the flow are sufficient for the description.
Two problems arise then. Knowing the viscosity of the fluid injected, the existence and the location of zones where the flow is stationary has to be determined a priori, without using a complete flow simulation on the detailed model. One also has to be able, with the same constraint, to choose the viscosity of the injected fluid so that a maximum number of layers has a stationary behaviour.
The method we propose allows to recognize the zones of a stratified reservoir behaving as a single layer in the hydrodynamic sense without using a fine simulation and without involving the vertical equilibrium hypotheses. It facilitates selection of the zones to be aggregated in the stratified reservoirs. It thus allows to take account of the dynamic and viscous effects while allowing very fast determination of the coarse layers in relation to the prior solutions.